First we note, by [1]}, that \(M\)  is a self-test module.
(i) The assertion follows from Theorem REF  by setting \(\operatorname{\mathcal {Z}}=\mathfrak {a})\) .
(ii) This is a special case of part (i).
(iii) We set \(\operatorname{\mathcal {Z}}=\lbrace \mathfrak {p}\in \operatorname{Spec}R\mid \operatorname{ht}(\mathfrak {p})>0\rbrace \) . Then, by Proposition REF (i) and our assumption, it follows that \(\Gamma _{\operatorname{\mathcal {Z}}}(\otimes _R^nM)=\otimes _R^nM)=0\) . Note by Proposition REF  that \(\operatorname{\textnormal {grade}}_R(\operatorname{\mathcal {Z}},R)>0\) . Hence, by Theorem REF (i), we may assume that \(\operatorname{\textnormal {grade}}_R(\operatorname{\mathcal {Z}},M)>0\) . Now the assertion follows from Theorem REF (ii).
